need help regarding stray cat
We are petsitting a friends cat. he is an active hunter and is outside a lot of the time. Since it has been cold he is inside more often. A week ago a black cat showed up on the property. It meows really long and loud, kind of whining. Our cat, his name is Max, ignores this cat and doesn't know what to think. He just looks at it and then walks away or pretends it isn't there. But recently the other cat has been sleeping in his chair and has following him inside to the bed he sleeps on. This cat basically pesters our cat and Max will come to us to get away and we chase the cat out of the house. He has no interest in this cat, hasn't fought it (to our knowledge) and doesn't hiss at it to get away. Max is fixed and it seems like this other cat is a female, but we haven't gotten that close to her to find out. She seems cute and like a pet of some sort, but we aren't sure. She doesn't look ratty, but with this cold weather we are worried she has no where to go. Since this isn't our house or our cat it isn't our decision to keep it around so we have avoided feeding it and haven't let it inside. We live in San Diego California so even during a cold winter here it still isn't that cold.
Any ideas on why this cat is pestering our cat or what to do?